3.2.40 \(\int \frac {b x^2+c x^4}{x^4} \, dx\) [140]

Optimal. Leaf size=10 \[ -\frac {b}{x}+c x \]

[Out]

-b/x+c*x

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} c x-\frac {b}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^4,x]

[Out]

-(b/x) + c*x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \frac {b x^2+c x^4}{x^4} \, dx &=\int \left (c+\frac {b}{x^2}\right ) \, dx\\ &=-\frac {b}{x}+c x\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {b}{x}+c x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^4,x]

[Out]

-(b/x) + c*x

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Maple [A]
time = 0.02, size = 11, normalized size = 1.10

method	result	size
default	\(-\frac {b}{x}+c x\)	\(11\)
risch	\(-\frac {b}{x}+c x\)	\(11\)
gosper	\(-\frac {-c \,x^{2}+b}{x}\)	\(14\)
norman	\(\frac {c \,x^{4}-b \,x^{2}}{x^{3}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-b/x+c*x

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Maxima [A]
time = 0.27, size = 10, normalized size = 1.00 \begin {gather*} c x - \frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^4,x, algorithm="maxima")

[Out]

c*x - b/x

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Fricas [A]
time = 0.32, size = 13, normalized size = 1.30 \begin {gather*} \frac {c x^{2} - b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^4,x, algorithm="fricas")

[Out]

(c*x^2 - b)/x

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Sympy [A]
time = 0.02, size = 5, normalized size = 0.50 \begin {gather*} - \frac {b}{x} + c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)/x**4,x)

[Out]

-b/x + c*x

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Giac [A]
time = 3.85, size = 10, normalized size = 1.00 \begin {gather*} c x - \frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^4,x, algorithm="giac")

[Out]

c*x - b/x

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Mupad [B]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} c\,x-\frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^4,x)

[Out]

c*x - b/x

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